# Recovering multiple fractional orders in time-fractional diffusion in an unknown medium

@article{Jin2021RecoveringMF, title={Recovering multiple fractional orders in time-fractional diffusion in an unknown medium}, author={Bangti Jin and Yavar Kian}, journal={Proceedings of the Royal Society A}, year={2021}, volume={477} }

In this work, we investigate an inverse problem of recovering multiple orders in a time-fractional diffusion model from the data observed at one single point on the boundary. We prove the unique recovery of the orders together with their weights, which does not require a full knowledge of the domain or medium properties, e.g. diffusion and potential coefficients, initial condition and source in the model. The proof is based on Laplace transform and asymptotic expansion. Furthermore, inspired by…

## 9 Citations

### Recovery of the Order of Derivation for Fractional Diffusion Equations in an Unknown Medium

- MathematicsSIAM Journal on Applied Mathematics
- 2022

In this work, we investigate the recovery of a parameter in a diffusion process given by the order of derivation in time for a class of diffusion type equations, including both classical and…

### Uniqueness of orders and parameters in multi-term time-fractional diffusion equations by inexact data

- Mathematics
- 2022

As the most signiﬁcant difference from parabolic equations, the asymptotic behavior of solutions to time-fractional evolution equations is dominated by the fractional orders, whose unique…

### Recovery of a Distributed Order Fractional Derivative in an Unknown Medium

- MathematicsArXiv
- 2022

In this work, we study an inverse problem of recovering information about the weight distribution in a distributed-order time-fractional diﬀusion model from the observation at one single point on the…

### Inverse source problem with a posteriori boundary measurement for fractional diffusion equations

- Mathematics
- 2022

. In this article we study inverse source problems for time-fractional diﬀusion equations from a posteriori boundary measurement. Using the memory eﬀect of these class of equations, we solve these…

### Recovering a Space-Dependent Source Term in the Fractional Diffusion Equation with the Riemann–Liouville Derivative

- MathematicsMathematics
- 2022

This research determines an unknown source term in the fractional diffusion equation with the Riemann–Liouville derivative. This problem is ill-posed. Conditional stability for the inverse source…

### Determining damping terms in fractional wave equations

- Mathematics
- 2022

This paper deals with the inverse problem of recovering an arbitrary number of fractional damping terms in a wave equation. We develop several approaches on uniqueness and reconstruction, some of…

### Equivalence of definitions of solutions for some class of fractional diffusion equations

- Mathematics
- 2021

We study the unique existence of weak solutions for initial boundary value problems associated with different class of fractional diffusion equations including variable order, distributed order and…

### Parameter Estimation for a Type of Fractional Diffusion Equation Based on Compact Difference Scheme

- MathematicsSymmetry
- 2022

Numerical solution and parameter estimation for a type of fractional diffusion equation are considered. Firstly, the symmetrical compact difference scheme is applied to solve the forward problem of…

### The enclosure method for the detection of variable order in fractional diffusion equations

- MathematicsInverse Problems and Imaging
- 2022

This paper is concerned with a new type of inverse obstacle problem governed by a variable-order time-fraction diffusion equation in a bounded domain. The unknown obstacle is a region where the space…

## References

SHOWING 1-10 OF 39 REFERENCES

### Recovery of the Order of Derivation for Fractional Diffusion Equations in an Unknown Medium

- MathematicsSIAM Journal on Applied Mathematics
- 2022

In this work, we investigate the recovery of a parameter in a diffusion process given by the order of derivation in time for a class of diffusion type equations, including both classical and…

### Simultaneous inversion of the potential term and the fractional orders in a multi-term time-fractional diffusion equation

- MathematicsInverse Problems
- 2021

In the present paper, we devote our effort to a nonlinear inverse problem for simultaneously recovering the potential function and the fractional orders in a multi-term time-fractional diffusion…

### Identifying a fractional order and a space source term in a time-fractional diffusion-wave equation simultaneously

- MathematicsInverse Problems
- 2019

This paper is devoted to identify simultaneously a fractional order and a space source term in a multi-dimensional time-fractional diffusion-wave equation by the final time measurement data. We prove…

### Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients

- MathematicsAppl. Math. Comput.
- 2015

### Two Fully Discrete Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data

- MathematicsSIAM J. Sci. Comput.
- 2016

Two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method are developed and establish error estimates optimal with respect to the regularity of problem data.

### Determination of order in fractional diffusion equation

- Mathematics
- 2013

We prove formulae of reconstructing the order of fractional derivative in time in the fractional diffusion equation by time history at one fixed spatial point. The proof is based on asymptotics of…

### Identification of time-varying source term in time-fractional diffusion equations

- MathematicsCommunications in Mathematical Sciences
- 2022

This paper is concerned with the inverse problem of determining the time and space dependent source term of diffusion equations with constant-order time-fractional derivative in (0, 2). We examine…

### Unique determination of fractional order and source term in a fractional diffusion equation from sparse boundary data

- MathematicsInverse Problems
- 2020

In this article, for a two dimensional fractional diffusion equation, we study an inverse problem for simultaneous restoration of the fractional order and the source term from the sparse boundary…

### Optimal Decay Estimates for Time-Fractional and Other NonLocal Subdiffusion Equations via Energy Methods

- MathematicsSIAM J. Math. Anal.
- 2015

We prove sharp estimates for the decay in time of solutions to a rather general class of nonlocal in time subdiffusion equations on a bounded domain subject to a homogeneous Dirichlet boundary…